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                <h1 id="title" titleSize="">
                    Group Action
                </h1>
            
            <h1 id="motivation--definition">Motivation &amp; Definition</h1>
<p><a href=https://zhaoshenzhai.github.io/mathwiki/group.md class="internalLink references" title="Group" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/group.md&#34;, {&#34;Date&#34;:&#34;2024-05-16T21:34:09-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-05-16T21:34:09-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-05-16T21:34:09-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Group&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/group&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-05-16T21:34:09-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-05-16T21:34:09-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-05-16T21:34:09-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Group&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/group&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/group.md&#34;, {&#34;Date&#34;:&#34;2024-05-16T21:34:09-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-05-16T21:34:09-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-05-16T21:34:09-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Group&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/group&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">Groups</a> capture $-$ <em>abstractly</em> $-$ the notion of a symmetry. But symmetries are also present in ‘concrete objects’, with different kinds of objects preserving different kinds of structure. The concept of a <em>group action</em> connects the two.</p>
<div class="env envDef" id=""><img class="icon noSelect listenDark" src="https://zhaoshenzhai.github.io/mathwiki/css/fa/definition.svg"><b class="envTitle">Definition. </b>Let $G$ be a group and fix a <a href=https://zhaoshenzhai.github.io/mathwiki/category.md class="internalLink references" title="Category" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/category.md&#34;, {&#34;Date&#34;:&#34;2024-05-22T21:52:00-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-05-22T21:52:00-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-05-22T21:52:00-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Category&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/category&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-05-22T21:52:00-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-05-22T21:52:00-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-05-22T21:52:00-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Category&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/category&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/category.md&#34;, {&#34;Date&#34;:&#34;2024-05-22T21:52:00-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-05-22T21:52:00-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-05-22T21:52:00-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Category&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/category&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">category</a> $C$. A <em>$G$-action</em> is a <a href=https://zhaoshenzhai.github.io/mathwiki/functor.md class="internalLink references" title="Functor" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/functor.md&#34;, {&#34;Date&#34;:&#34;2024-05-28T16:05:23-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-05-28T16:05:23-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-05-28T16:05:23-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Functor&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/functor&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-05-28T16:05:23-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-05-28T16:05:23-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-05-28T16:05:23-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Functor&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/functor&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/functor.md&#34;, {&#34;Date&#34;:&#34;2024-05-28T16:05:23-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-05-28T16:05:23-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-05-28T16:05:23-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Functor&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/functor&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">functor</a> $\cat{B}G\to C$, which specifies an object $X\in C$ and a homomorphism $\phi:G\to\Aut X$.</div>

<p>Thus a group action is specified by the data of a tuple $\tpl{X,\phi}$, which we usually write as $\phi:G\act X$, and write $\phi_g(x)\coloneqq\phi(g)(x)$. The object $X$ is the ‘concrete object’, and $\phi$ encodes how $G$ manifests itself as $\Aut X$, the symmetries of $X$. Note, however, that $\phi$ need not be injective; we say that the action is <em>faithful</em> if it is, so that $G\into\Aut X$. Otherwise, $Z\coloneqq\ker\phi$ is non-trivial.</p>
<h2 id="basic-notions">Basic Notions</h2>
<p>For $C$ a concrete category (e.g. $\catset,\catvect[k],\cattop,\dots$), there are some important constructions.</p>
<ul>
<li>The <em>orbit</em> of a fixed $x\in X$ is the set $Gx\coloneqq\l\{gx\in X\st g\in G\r\}$ of <em>$G$-translates</em>, and the <em>stabilizer</em> of $x$ is $G_x\coloneqq\l\{g\in G\st gx=x\r\}$. Note that $Z=\ker\phi=\bigcap_{x\in X}G_x$, so it is the set of <em>fixed points</em> of $\phi$.</li>
<li>Dually, the <em>$g$-translate</em> of a fixed $g\in G$ is the set $gX\coloneqq\l\{gx\st x\in X\r\}$, and the <em>fixed points</em> of $g$ is $X^g\coloneqq\l\{x\in X\st gx=x\r\}$.</li>
</ul>
<p>Note that the orbits form a partition $X=\bigsqcup_iGx_i$ and $G\act Gx_i$ for each $i$ independent of each other, so we will sometimes just focus on actions of $G$ on a single orbit. In this case, when $G\act X$ has a single orbit, we say that $G$ acts <em>transitively</em> on $X$; that is, for every $x,y\in X$, we have $y=gx$ for some $g\in X$.</p>
<br>
<p>  A stronger criteria than faithfulness is when $G_x$ is trivial for <em>all</em> $x\in X$, in which case we say that $\phi$ is <em>free</em>. While the former has no <em>global</em> fixed points ($\fa g\in G(\fa x\in X:gx=x)\rightarrow g=e$), free actions admit no <em>local</em> ones ($\fa g\in G(\ex x\in X:gx=x)\rightarrow g=e$).</p>
<h2 id="the-category-cg">The Category $C^G$</h2>
<p>Since group actions are functors, it makes sense to consider <a href=https://zhaoshenzhai.github.io/mathwiki/natural_transformation.md class="internalLink references" title="Natural Transformation" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/natural_transformation.md&#34;, {&#34;Date&#34;:&#34;2024-05-29T13:41:19-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-05-29T13:41:19-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-05-29T13:41:19-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Natural Transformation&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/natural_transformation&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-05-29T13:41:19-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-05-29T13:41:19-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-05-29T13:41:19-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Natural Transformation&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/natural_transformation&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/natural_transformation.md&#34;, {&#34;Date&#34;:&#34;2024-05-29T13:41:19-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-05-29T13:41:19-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-05-29T13:41:19-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Natural Transformation&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/natural_transformation&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">natural transformations</a> between two actions $\phi:G\act X$ and $\psi:G\act Y$ $-$ called <em>$G$-maps</em> $-$ which are the morphisms in the category $C^G$ of $G$-actions on $C$.</p>
<br>
<h1 id="actions-on-catset">Actions on $\catset$</h1>
<p>Taking $C\coloneqq\catset$ recovers the usual notion of a group acting on a set, in which case $\Aut X$ are just the permutations of $X$. Some important examples include:</p>
<ul>
<li>The <em>left-multiplication</em> action of $G$ on $X\coloneqq G$, given by $\phi_g(x)\coloneqq gx$. That this action is faithful is <a href=https://zhaoshenzhai.github.io/mathwiki/cayleys_theorem.md class="internalLink properties" title="Cayley’s Theorem" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/cayleys_theorem.md&#34;, {&#34;Date&#34;:&#34;2024-06-20T19:25:15-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-20T19:25:15-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-20T19:25:15-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Cayley’s Theorem&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/cayleys_theorem&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-06-20T19:25:15-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-20T19:25:15-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-20T19:25:15-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Cayley’s Theorem&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/cayleys_theorem&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/cayleys_theorem.md&#34;, {&#34;Date&#34;:&#34;2024-06-20T19:25:15-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-20T19:25:15-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-20T19:25:15-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Cayley’s Theorem&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/cayleys_theorem&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">Cayley’s Theorem</a>. Moreover, it turns out<a href=https://zhaoshenzhai.github.io/mathwiki/orbit_stabilizer_theorem.md class="internalLink properties dag" title="Orbit Stabilizer Theorem" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/orbit_stabilizer_theorem.md&#34;, {&#34;Date&#34;:&#34;2024-06-03T17:04:49-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-03T17:04:49-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-03T17:04:49-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Orbit Stabilizer Theorem&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/orbit_stabilizer_theorem&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-06-03T17:04:49-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-03T17:04:49-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-03T17:04:49-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Orbit Stabilizer Theorem&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/orbit_stabilizer_theorem&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/orbit_stabilizer_theorem.md&#34;, {&#34;Date&#34;:&#34;2024-06-03T17:04:49-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-03T17:04:49-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-03T17:04:49-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Orbit Stabilizer Theorem&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/orbit_stabilizer_theorem&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">$^\dagger$</a> that every action is the coproduct of certain left-multiplication actions of $G$.</li>
<li>The <em><a href=https://zhaoshenzhai.github.io/mathwiki/conjugation_action.md class="internalLink examples" title="Conjugation Action" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/conjugation_action.md&#34;, {&#34;Date&#34;:&#34;2024-06-21T20:09:18-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-21T20:09:18-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-21T20:09:18-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Conjugation Action&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/conjugation_action&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-06-21T20:09:18-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-21T20:09:18-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-21T20:09:18-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Conjugation Action&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/conjugation_action&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/conjugation_action.md&#34;, {&#34;Date&#34;:&#34;2024-06-21T20:09:18-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-21T20:09:18-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-21T20:09:18-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Conjugation Action&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/conjugation_action&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">conjugation</a></em> action of $G$ on $X\coloneqq G$, given by $\phi_g(x)\coloneqq gxg^{-1}$. The <em>center</em> of $G$ is the fixed points $Z(G)$, whose combinatorics is governed by the <a href=https://zhaoshenzhai.github.io/mathwiki/the_class_equation.md class="internalLink properties" title="The Class Equation" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/the_class_equation.md&#34;, {&#34;Date&#34;:&#34;2024-06-21T14:50:24-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-21T14:50:24-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-21T14:50:24-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;The Class Equation&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/the_class_equation&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-06-21T14:50:24-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-21T14:50:24-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-21T14:50:24-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;The Class Equation&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/the_class_equation&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/the_class_equation.md&#34;, {&#34;Date&#34;:&#34;2024-06-21T14:50:24-04:00&#34;,&#34;Lastmod&#34;:&#34;2024-06-21T14:50:24-04:00&#34;,&#34;PublishDate&#34;:&#34;2024-06-21T14:50:24-04:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;The Class Equation&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/the_class_equation&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">Class Equation</a>.</li>
</ul>
<h1 id="generalizations">Generalizations</h1>
<p>More generally, we can $-$ in the definition $-$ replace $G$ with a <a href=https://zhaoshenzhai.github.io/mathwiki/group_object.md class="internalLink references ghostLink" title="group object" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/group_object.md&#34;, &#34;nopPage&#34;);" onmouseleave="clearPreviewSide(&#34;nopPage&#34;);" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/group_object.md&#34;, &#34;nopPage&#34;);">group object</a> in $C$, giving us <a href=https://zhaoshenzhai.github.io/mathwiki/continuous_group_action.md class="internalLink types ghostLink" title="continuous group actions" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/continuous_group_action.md&#34;, &#34;nopPage&#34;);" onmouseleave="clearPreviewSide(&#34;nopPage&#34;);" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/continuous_group_action.md&#34;, &#34;nopPage&#34;);">continuous group actions</a>, <a href=https://zhaoshenzhai.github.io/mathwiki/lie_group_action.md class="internalLink types ghostLink" title="Lie group actions" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/lie_group_action.md&#34;, &#34;nopPage&#34;);" onmouseleave="clearPreviewSide(&#34;nopPage&#34;);" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/lie_group_action.md&#34;, &#34;nopPage&#34;);">Lie group actions</a>, etc; the definitions above carry forth with obvious modifications.</p>
<br>
<p>  Representation Theory, GGT, etc&hellip;</p>


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                May 22, 2024 | Zhaoshen Zhai

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